Hi, Alice --
Thanks for writing to T2T. Students do need first and foremost to understand
what multiplication means. Concrete materials is a good way to do that. They
need to see multiplication as a more efficient way to add when the groups are
all the same size. Without understanding what multiplication does, memorizing
algorithms does them no good. They won't be able to apply the procedures to
anything meaningful, in a test or otherwise.
With that said, they also need to develop fluency with computation as soon as
they do have the concepts under their belts. Otherwise they won't be able to
apply that understanding in an efficient way (in a test or otherwise). They
do need to learn number facts, develop number and operation sense, and master
some efficient algorithm that makes sense to them and that they have
confidence in.
For some students the lattice method is just the thing. While it's not
terribly intuitive, this ancient method does have sound rationale behind it,
and many children are able to apply it very reliably.
There are other algorithms that work better for other children. My own
opinion is that children who are taught several different ones, and are given
enough practice with them, will discover the procedure that makes sense to
them and that they can use accurately. If it makes sense to them, they are
much more likely to remember it. If memory fails at some point, they can
usually re-construct a method that they understand.
The Everyday Mathematics program uses this approach. You can learn more about
the algorithms they offer children at
http://instruction.aaps.k12.mi.us/EM_parent_hdbk/algorithms.html
The Partial Products method makes sense to many kids. Once they understand
it, they make an easy transition to the more traditional, more efficient
method. It takes advantage of, and improves, their number sense, and paves
the way for algebra.
I hope this is helpful. Please write again if you have more questions.
-Claire, for the T2T service